There are a number of speech-form clusters in the world, that is, genetically related languages which are so structurally similar that they are said to be "dialects" of a language – e.g. Saami, Shona, Somali, Luhya, Chinese, Arabic, Kurdish, Quechua and Mongolian. You might say that the various Saami languages are versions of Saami, or they are dialects of Saami, but there is no standard reference form that one could actually write down (likewise Quechua, Luhya and Kurdish). Because of the writing system, there is a standard thing that can be identified as Chinese – and nobody speaks that language (it's a written form, not a spoken form).
If by "Arabic" you mean either "Modern Standard Arabic" or "Classical Arabic", which is not a first language anymore, then it is a stretch to say that any modern spoken dialect is a "version" of the classical language, just as it is a stretch to say that any modern Chinese dialect is a "version" of classical Chinese or Khalkha Mongolian is a "version" of Classical Mongolian (idem classical Tibetan and modern Tibetan, and so on). The "Arabic versions" of Lebanon, Syria, Iraq, Egypt, Morocco, Nigeria and so on have a clear historical connection to Classical Arabic that is similar to the connection between the individual Romance languages and Latin, or the various Chinese dialects and an earlier common version of Chinese. They all derive historically from a common language, just as the North Germanic, Saami and Mongolian languages do. The primary reason that most "versions of Arabic" are called Arabic is political (and the reason why Maltese is not generally called Arabic is likewise political, though it is clearly not as similar to Classical Arabic as Spoken Levantine Arabic is). In other words, saying that Lebanese is "a version of Arabic" does not clarify anything, and you can't decide whether it is "a version", unless you have a clear and well-justified terminological alternative. (If you were speaking of Chaldean Neo-Aramaic in Iraq, that could not be considered a "version" of Arabic). It is sensible to ask about the relationship between two genetically-related languages, it's just that "version" does not impart any clarity to the discussion.
There is a significant similarity between Classical Arabic and Spoken Lebanese Arabic in the realm of grammar, and a huge similarity between Lebanese, Syrian, Iraqi, Egyptian and so on. For example, the patterns of verb inflection in the modern languages are extremely similar, and recognizable related to Classical Arabic – though there are a number of things in Classical Arabic that have disappeared (for instance, jussive and subjunctive inflections). I think the 1sg perfective suffix is /t/ in all modern Arabic varieties, clearly related to Classical /tu/.
The way that phonologists think about "the phonology" of a language, phonology is primarily about the rules which manipulate roots and affixes, so that in some dialects, /katabt/ is pronounced [katábit] (or, elsewhere, /katab/ it pronounced [kítab]). This system of rules is complex enough to be an area of significant interest for linguists and has been reasonably well studied. I would agree that there is not much similarity between Classical Arabic and the modern dialects, but it is easy to see how the phonologies of the modern dialects arose historically. For instance, in Classical Arabic the roots /dʕw/ and /rmy/ end with different underlying glides; in the modern languages, the glide is gone and instead you get epenthetic e:, i:; the inflection of roots like /mdd/ has changes drastically so that madadtu ~ maddat gave way to madde:t ~ maddat. It would be a wonderful project to line up the relevant properties and compose a model of relatedness based on such shared changes from the classical language: this has been done somewhat, and research on the topic will continue for many years.
If some other Arabic speakers think you are Israeli, that probably reflects the fact that the varieties of Arabic spoken along the eastern coast of the Mediterranean are extremely similar. It would be interesting, but quite impractical, to conduct a well-controlled and extensive survey of perceived similarity among varieties of Arabic. Factoring in language-external comparisons such as Aramaic-Arabic, Berber-Arabic is just that much more difficult: not impossible, but difficult. Please note that this just looks at large-scale subjective judgments of "similarity"; another approach is to develop a metric of objective grammatical and lexical similarity.